Arguments that a bubble exists in the US Treasury market invariably reference the existing historically low yield environment.Β What is missing from many of these arguments is consideration of WHY many investors incorporate bonds into their portfolios in the first place.

For many investors, bonds are used to diversify the equity exposures in their portfolios, in essence providing a degree of insurance against large market dislocations^{1}.Β For these investors, it is not as simple as saying that bonds are expensive relative to past valuations and call it a day.Β Instead, we must evaluate the potential cost (premium in insurance terms) of employing bonds against the cost of other forms of portfolio insurance.

The obvious comparable instruments for protecting an equity portfolio are put options.Β For investors that use bonds as a type of insurance, we can use current option prices as another measure of just how costly bonds are.

This post will show that depending on equity market environment, it is possible for higher duration bond portfolios to add value as a diversification tool even in a rising interest rate environment.

The following relationship approximates the t-month ahead yield, y_t, that would lead to the bond approach returning the same amount as the option approach over a t-month period for a given equity return r_(0,t).

The notation in the above equation is as follows:

- D is the duration of the bond portfolio used to hedge equity risk
- y_0 is the current yield on the bond portfolio
- w is the allocation to the equity portfolio
- P is the price of a put option for a given maturity, implied volatility, risk-free rate, stock price and strike price
- k is the strike price as a percentage of the current stock price

To illustrate the type of analysis that can be conducted using this relationship, we can consider an example.Β Assume an investor, invested in the SPDR S&P 500 ETF (βSPYβ), determines that a 30% holding of 10-year duration Treasuries long-term provide the same protection as continuously rolling over 3-month put options struck 2.5% out of the money^{2}.Β Therefore, we have:

- D = 10
- y_0 = 1.76% (10-year constant maturity yield)
- w = 0.7
- P = 2.76 where 14.46 is used at the implied volatility (CBOE S&P 500 3-Month Volatility Index), 0.25 (three months) is used as the time to maturity, 157.45 is used as the current SPY value, and 0.09% is used as the risk-free rate (3-month T-Bill rate)
- k = 0.975

The following graph shows the breakeven bond yield given various 3-month SPY returns.

The blue line is the breakeven bond yield and the red is the current 10-year constant maturity US Treasury yield.Β If the actual yield in three months is below the blue line given the actual SPY return, then the bond hedge will outperform the option hedge.Β Likewise, if the actual yield in three months is above the blue line given the actual SPY return, then the bond hedge will underperform the option hedge.

For example, if SPY is flat over the next three months, then the bond hedge will outperform the option hedge as long as yields remain below 2.52%, a 43% rise over current rates.Β To put this into context, the largest quarter-over-quarter increase in 10-year rates was approximately 21% in the first quarter of 2009.

We see that option hedge only becomes cost effective relative to bonds for large SPY moves.Β This is especially relevant given the current low VXV levels (i.e. options are relatively inexpensive compared to the recent past).Β If VXV rises, then the ability of bonds to outperform options as an equity hedge will increase further, even if yields rise.

Consider an investor who was invested in the S&P 500 in 2007 through SPY.Β To insure his portfolio, the investor buys three-month put options each quarter that are 2.5% out of the money.Β We assume that the investor is able to purchase the options at an implied volatility equal to VXV.Β Such a strategy ensures a maximum 1-year portfolio drawdown of 10.0% before considering the cost of the option.

Unfortunately, buying volatility through options becomes most expensive when the protection is most needed.Β Taking into account option cost, the investor would have experienced a maximum drawdown of 29.9%Β During this same period, SPY experienced a maximum drawdown of 43.9%.

To achieve a similar degree of drawdown reduction using 10-year duration US Treasuries, the investor would have needed to allocate 30% of his portfolio to bonds.Β Using this approach, the maximum drawdown experienced during the global credit crisis would have been 28.5%.

Note that we used quarterly price data for this exercise and as a result drawdown statistics will be different if we used daily price and marked the options to market throughout the holding period.

^{1Β }This is not insurance in the purest sense of the word since it relies on the assumption that equity/bond correlations will be negative in market crashes, an assumption that we cannot be 100% sure of going forward.

^{2Β }We came up with these numbers through a quick comparison of the two strategies during the global credit crisis.Β Obviously, investors would have to perform much deeper analysis to determine if they could construct a bond hedge that gave them the same comfort as an option hedge and what these two hedges look like.

^{3Β }By using VXV throughout this example to price out of the money put options, we are implicitly making the assumption that there is no skew/smile in option volatility.Β Note, however, that building in skew/smile would have made the out of the money put options more expensive and increased the relative attractiveness of utilizing bonds.

^{4Β }Using 10-year rates for a 10-year duration portfolio is not completely accurate, the duration on 10-year bonds will be less than 10-years.Β However, the analysis remains directionally accurate.

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